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\name{RMcov}
\alias{RMcov}
\alias{RMcov}
\alias{RMCOV_X}
\title{Non-stationary covariance model corresponding to a variogram model}
\description{
This function generalizes the well-known non-stationary covariance
function \eqn{2\min\{x,y\}} of the Brownian motion with variogram
\eqn{\gamma(x,y) = |x-y|}, \eqn{x,y\ge 0}
to arbitrary variogram models any spatial processes of any dimension
and multivariability.
Furthermore, the
standard condition for the Brownian motion \eqn{W} is that
variance equals \eqn{0} at the origin,
i.e., \eqn{W(x) =^d Z(x) -Z(0)} for any zero mean Gaussian process
\eqn{Z} with variogram \eqn{\gamma(x,y) = |x-y|} is replaced by
\eqn{W(x) = Z(x) -\sum_{i=1}^n a_i Z(x_i)} with \eqn{\sum_{i=1}^n a_i
= 1}.
For a given variogram \eqn{\gamma}, \eqn{a_i} and \eqn{x_i}, the model
equals
\eqn{C(x, y) = \sum_{i=1}^n a_i (\gamma(x, x_i) + \gamma(x_i, y)) -
\gamma(x, y) - \sum_{i=1}^n \sum_{j=1}^n a_i a_j \gamma(x_i, y_i)
}
}
\usage{
RMcov(gamma, x, y=NULL, z=NULL, T=NULL, grid, a,
var, scale, Aniso, proj, raw, norm)
}
\arguments{
\item{gamma}{a variogram model. Possibly multivariate.}
\item{x,y,z,T,grid}{
The usual arguments as in \command{\link{RFsimulate}} to define the
locations where the covariates are given.
Additional \code{x} might be set to one of the values
\code{"origin"}, \code{"center"}, \code{"extremals"}, or \code{"all"}.
If \code{x} is not given, \code{x} is set to \code{"origin"}.
}
\item{a}{vector of weights. The length of \code{a} must equal the
number of points given by \code{x}, \code{y}, \code{z} and \code{T}.
The values of \code{a} must sum up to \eqn{1}.
If \code{a} is not given, equals weights are used.
}
\item{var,scale,Aniso,proj}{optional arguments; same meaning for any
\command{\link{RMmodel}}. If not passed, the above covariance function
remains unmodified.}
\item{raw}{\argRaw}
\item{norm}{\argNorm}
}
%\details{}
\value{
\command{\link{RMcov}} returns an object of class \code{\link[=RMmodel-class]{RMmodel}}
}
\author{Martin Schlather, \email{schlather@math.uni-mannheim.de}
}
\seealso{
\command{\link{RMmodel}},
\command{\link{RFsimulate}},
\command{\link{RFfit}}.
}
\keyword{spatial}
\keyword{models}
\examples{\dontshow{StartExample()}
RFoptions(seed=0) ## *ANY* simulation will have the random seed 0; set
## RFoptions(seed=NA) to make them all random again
bm <- RMfbm(alpha=1)
plot(bm)
x <- seq(0, 6, if (interactive()) 0.125 else 3)
plot(RFsimulate(bm, x))
## standardizing with the random variable at the origin
z1 <- RFsimulate(RMcov(bm), x)
plot(z1)
z1 <- as.vector(z1)
zero <- which(abs(x) == 0)
stopifnot(abs(z1[zero]) < 1e-13)
## standardizing with the random variable at the center of the interval
z2 <- RFsimulate(RMcov(bm, "center"), x)
plot(z2)
z2 <- as.vector(z2)
stopifnot(abs(z2[(length(z2) + 1) / 2]) < 1e-13)
## standardizing with the random variables at the end points of the interval
z3 <- RFsimulate(RMcov(bm, "extremals"), x)
plot(z3)
z3 <- as.vector(z3)
stopifnot(abs(z3[1] + z3[length(z3)]) < 1e-13)
\dontshow{FinalizeExample()}
}
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